If they didn't provide you with a state-by-state breakdown (and obviously they didn't, otherwise you wouldn't be asking), then I would guess that no state returns need to be filed. Although you should probably call that phone # and ask them to verify that. (They should have sent you a "State Schedule" if you need to file state returns.)

Might this mean that since it is a return of capital, no state taxes are owed anywhere?

Unfortunately, just because no taxes are due, doesn't necessarily mean that (some) state returns don't need to be filed. Not only that, but just because the dividends are return of capital doesn't mean that no taxes are due! See the FAQ that reads What is Unrelated Business Taxable Income (UBTI)?. (I believe UBTI applies only to tax-exempt accounts, however, such as IRAs, so that's probably not relevant here.) And, lastly, just because all the dividends are ROC, and the UBTI is not applicable, doesn't mean no taxes are owed. Your personal taxes are based on other things, such as net income, 1231 & 1245 gains, capital gains, etc.

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